Incompleteness theoremand its frontier

Summary

As you know well, we celebrate 70th year of Gödel's “Incompleteness theorem” which he proved at the age of 25. This revolutionary theorem changed the way mathematicians think of mathematics drastically. Today, I would like to describe this theorem, its effects and expound on this subject.

Gödel's incompleteness theorem can be stated as follows.

1. 1. Let T be a consistent axiomatic theory like set theory or the theory of analysis where Peano's arithmetic (denoted simply as arithmetic from now on) is included. Then T does not prove its consistency.

2. 2. The consistency of T can be expressed as a sentence in arithmetic, therefore there exists an arithmetical sentence which cannot be proved in T.

This seemingly simple theorem changed our view of mathematics completely. Before this theorem, mathematicians believed that every problem in arithmetic could be solved by some stronger theory, e.g., set theory. But the Incompleteness theorem tells us that whatever stronger theory we use, there exists a true arithmetical sentence which cannot be proved in the theory.

After the Incompleteness theorem, the consistency statement becomes a landmark for the boundary of provable statements in the theory. Whenever we wish to show that the theory T is strictly stronger than the theory T, we first try to show that the theory T proves the consistency statement of T. In this way we can show that the theory of analysis is strictly stronger than the theory of arithmetic and that set theory is strictly stronger than the theory of analysis.

I would now like to speak of the impact that this theorem had on Hilbert's program. Hilbert was a genius to think of problems in a general setting, to find the essence of the problem and solve it. So, he was a leader of the movement of abstract systematic development and axiomatization in the 20th century mathematics. Cantor's set theory gave the ideal framework for this movement. Hilbert believed Cantor had created a new paradise for mathematicians. So, paradoxes of Cantor's set theory came as a great shock to him. Hilbert tried to save themodernmathematics and proposed the following program.